Question: Solve for $x$ and $y$ using elimination. $\begin{align*}5x-3y &= 1 \\ 7x-y &= 1\end{align*}$
Explanation: We can eliminate $y$ when its corresponding coefficients are negative inverses. Recalling our knowledge of least common multiples, multiply the top equation by $-1$ and the bottom equation by $3$ $\begin{align*}-5x+3y &= -1\\ 21x-3y &= 3\end{align*}$ Add the top and bottom equations. $16x = 2$ Divide both sides by $16$ and reduce as necessary. $x = \dfrac{1}{8}$ Substitute $\dfrac{1}{8}$ for $x$ in the top equation. $5( \dfrac{1}{8})-3y = 1$ $\dfrac{5}{8}-3y = 1$ $-3y = \dfrac{3}{8}$ $y = -\dfrac{1}{8}$ The solution is $\enspace x = \dfrac{1}{8}, \enspace y = -\dfrac{1}{8}$.